on ballico-hefez curves and associated supersingular surfaces

SUPERSINGULAR SURFACES

HOANG THANH HOAI AND ICHIRO SHIMADA

Abstract. Letpbe a prime integer, andqa power ofp. The Ballico-Hefez curve is a non-reflexive nodal rational plane curve of degreeq+ 1 in charac- teristicp. We investigate its automorphism group and defining equation. We also prove that the surface obtained as the cyclic cover of the projective plane branched along the Ballico-Hefez curve is unirational, and hence is supersingu- lar. As an application, we obtain a new projective model of the supersingular K3 surface with Artin invariant 1 in characteristic 3 and 5.

1. Introduction

We work over an algebraically closed fieldkof positive characteristicp >0. Let q=p^ν be a power of p.

In positive characteristics, algebraic varieties often possess interesting properties that are not observed in characteristic zero. One of those properties is the failure of reflexivity. In [4], Ballico and Hefez classified irreducible plane curvesX of degree q+ 1 such that the natural morphism from the conormal variety C(X) of X to the dual curve X^∨ has inseparable degree q. The Ballico-Hefez curve in the title of this note is one of the curves that appear in their classification. It is defined in Fukasawa, Homma and Kim [8] as follows.

Definition 1.1. TheBallico-Hefez curveis the image of the morphismφ:P¹→P² defined by

[s:t]7→[s^q+1:t^q+1:st^q+s^qt].

Theorem 1.2(Ballico and Hefez [4], Fukasawa, Homma and Kim [8]). (1)LetBbe the Ballico-Hefez curve. ThenB is a curve of degreeq+ 1with(q²−q)/2ordinary nodes, the dual curveB^∨ is of degree2, and the natural morphismC(B)→B^∨ has inseparable degreeq.

(2)LetX ⊂P²be an irreducible singular curve of degreeq+ 1such that the dual curveX^∨ is of degree >1 and the natural morphismC(X)→X^∨ has inseparable degreeq. ThenX is projectively isomorphic to the Ballico-Hefez curve.

Recently, geometry and arithmetic of the Ballico-Hefez curve have been inves- tigated by Fukasawa, Homma and Kim [8] and Fukasawa [7] from various points of view, including coding theory and Galois points. As is pointed out in [8], the Ballico-Hefez curve has many properties in common with the Hermitian curve; that

2000Mathematics Subject Classification. primary 14H45, secondary 14J25.

Partially supported by JSPS Grant-in-Aid for Challenging Exploratory Research No.23654012 and JSPS Grants-in-Aid for Scientific Research (C) No.25400042 .

1

is, the Fermat curve of degreeq+1, which also appears in the classification of Ballico and Hefez [4]. In fact, we can easily see that the image of the line

x0+x1+x2= 0 inP²by the morphism P²→P² given by

[x0:x1:x2]7→[x^q+1₀ :x^q+1₁ :x^q+1₂ ]

is projectively isomorphic to the Ballico-Hefez curve. Hence, up to linear transfor- mation of coordinates, the Ballico-Hefez curve is defined by an equation

x

1 q+1

0 +x

1 q+1

1 +x

1 q+1

2 = 0 in the style of “Coxeter curves” (see Griffith [9]).

In this note, we prove the the following:

Proposition 1.3. Let B be the Ballico-Hefez curve. Then the group Aut(B) :={g∈PGL3(k) | g(B) =B } of projective automorphisms ofB⊂P² is isomorphic to PGL2(Fq).

Proposition 1.4. The Ballico-Hefez curve is defined by the following equations:

• Whenp= 2,

x^q₀x₁+x₀x^q₁+x^q+1₂ +

ν∑−1 i=0

x²₀ⁱx²₁ⁱx^q+1₂ ⁻²ⁱ⁺¹= 0, whereq= 2^ν.

• Whenpis odd,

2(x^q₀x1+x0x^q₁)−x^q+1₂ −(x²₂−4x1x0)^q+12 = 0.

Remark1.5. In fact, the defining equation forp= 2 has been obtained by Fukasawa in an apparently different form (see Remark 3 of [6]).

Another property of algebraic varieties peculiar to positive characteristics is the failure of L¨uroth’s theorem for surfaces; a non-rational surface can be unirational in positive characteristics. A famous example of this phenomenon is the Fermat surface of degreeq+1. Shioda [18] and Shioda-Katsura [19] showed that the Fermat surfaceF of degreeq+1 is unirational (see also [16] for another proof). This surface F is obtained as the cyclic cover ofP²with degreeq+ 1 branched along the Fermat curve of degree q+ 1, and hence, for any divisord ofq+ 1, the cyclic cover ofP² with degreedbranched along the Fermat curve of degreeq+ 1 is also unirational.

We prove an analogue of this result for the Ballico-Hefez curve. Letdbe a divisor ofq+ 1 larger than 1. Note thatdis prime top.

Proposition 1.6. Let γ : S_d → P² be the cyclic covering of P² with degree d branched along the Ballico-Hefez curve. Then there exists a dominant rational map P²· · · →S_d of degree 2q with inseparable degreeq.

Note thatSd is not rational except for the case (d, q+ 1) = (3,3) or (2,4).

A smooth surface X is said to besupersingular (in the sense of Shioda) if the secondl-adic cohomology groupH²(X) ofX is generated by the classes of curves.

Shioda [18] proved that every smooth unirational surface is supersingular. Hence we obtain the following:

Corollary 1.7. Letρ: ˜Sd →Sd be the minimal resolution ofSd. Then the surface S˜d is supersingular.

We present a finite set of curves on ˜Sd whose classes spanH²( ˜Sd). For a point P of P¹, let l_P ⊂ P² denote the line tangent at φ(P) ∈ B to the branch of B corresponding to P. It was shown in [8] that, if P is an Fq²-rational point of P¹, thenl_P andB intersect only at φ(P), and hence the strict transform ofl_P by the composite ˜S_d →S_d→P² is a union ofdrational curvesl_P⁽⁰⁾, . . . , l^(d_P⁻¹⁾.

Proposition 1.8. The cohomology group H²( ˜S_d) is generated by the classes of the following rational curves on S˜_d; the irreducible components of the exceptional divisor of the resolution ρ : ˜S_d → S_d and the rational curves l_P⁽ⁱ⁾, where P runs through the set P¹(Fq²)of Fq²-rational points of P¹ andi= 0, . . . , d−1.

Note that, when (d, q+ 1) = (4,4) and (2,6), the surface ˜Sd is aK3 surface. In these cases, we can prove that the classes of rational curves given in Proposition 1.8 generate the N´eron-Severi lattice NS( ˜Sd) of ˜Sd, and that the discriminant of NS( ˜Sd) is−p². Using this fact and the result of Ogus [13, 14] and Rudakov-Shafarevic [15]

on the uniqueness of a supersingular K3 surface with Artin invariant 1, we prove the following:

Proposition 1.9. (1) If p=q = 3, then S˜4 is isomorphic to the Fermat quartic surface

w⁴+x⁴+y⁴+z⁴= 0.

(2)If p=q= 5, thenS˜2 is isomorphic to the Fermat sextic double plane w²=x⁶+y⁶+z⁶.

Recently, many studies on these supersingularK3 surfaces with Artin invariant 1 in characteristics 3 and 5 have been carried out. See [10, 12] for characteristic 3 case, and [11, 17] for characteristic 5 case.

Thanks are due to Masaaki Homma and Satoru Fukasawa for their comments.

We also thank the referee for his/her suggestion on the first version of this paper.

2. Basic properties of the Ballico-Hefez curve

We recall some properties of the Ballico-Hefez curveB. See Fukasawa, Homma and Kim [8] for the proofs.

It is easy to see that the morphismφ:P¹→P² is birational onto its image B, and that the degree of the plane curveB isq+ 1. The singular locus Sing(B) ofB consists of (q²−q)/2 ordinary nodes, and we have

φ⁻¹(Sing(B)) =P¹(Fq²)\P¹(Fq).

In particular, the singular locus Sing(S_d) of S_d consists of (q²−q)/2 ordinary rational double points of type A_d−1. Therefore, by Artin [1, 2], the surfaceS_d is not rational if (d, q+ 1)̸= (3,3),(2,4).

Lett be the affine coordinate of P¹ obtained from [s:t] by puttings= 1, and let (x, y) be the affine coordinates ofP² such that [x₀ :x₁:x₂] = [1 :x:y]. Then the morphismφ:P¹→P² is given by

t7→(t^q+1, t^q+t).

For a pointP = [1 :t] of P¹, the linelP is defined by x−t^qy+t^2q = 0.

Suppose thatP /∈P¹(Fq²). Then l_P intersects B at φ(P) = (t^q+1, t^q +t) with multiplicity q and at the point (t^q2+q, t^q2 +t^q) ̸= φ(P) with multiplicity 1. In particular, we havel_P∩Sing(B) =∅.

Suppose thatP ∈P¹(Fq²)\P¹(Fq). Thenl_P intersectsB at the nodeφ(P) ofB with multiplicityq+ 1. More precisely,l_P intersects the branch ofB corresponding toP with multiplicityq, and the other branch transversely.

Suppose thatP ∈P¹(Fq). Thenφ(P) is a smooth point ofB, andlP intersects B atφ(P) with multiplicityq+ 1. In particular, we havelP∩Sing(B) =∅.

Combining these facts, we see thatφ(P¹(Fq)) coincides with the set of smooth inflection points ofB. (See [8] for the definition of inflection points.)

3. Proof of Proposition 1.3

We denote by φ_B :P¹→B the birational morphismt7→(t^q+1, t^q +t) fromP¹ toB. We identify Aut(P¹) with PGL₂(k) by letting PGL₂(k) act onP¹by

[s:t]7→[as+bt:cs+dt] for

[ a b c d

]

∈PGL2(k).

Then PGL2(Fq) is the subgroup of PGL2(k) consisting of elements that leave the set P¹(Fq) invariant. SinceφBis birational, the projective automorphism group Aut(B) ofB acts onP¹ viaφB. The subsetφB(P¹(Fq)) of B is projectively characterized as the set of smooth inflection points ofB, and we haveP¹(Fq) =φ⁻_B¹(φB(P¹(Fq))).

Hence Aut(B) is contained in the subgroup PGL2(Fq) of PGL2(k). Thus, in order to prove Proposition 1.3, it is enough to show that every element

g:=

[ a b c d

]

with a, b, c, d∈Fq

of PGL₂(Fq) is coming from the action of an element of Aut(B). We put

˜ g:=



 a² b² ab c² d² cd 2ac 2bd ad+bc



,

and let the matrix ˜g act on P² by the left multiplication on the column vector

t[x0:x1:x2]. Then we have

φ◦g= ˜g◦φ,

because we have λ^q =λfor λ=a, b, c, d ∈Fq. Therefore g 7→˜g gives an isomor- phism from PGL₂(Fq) to Aut(B).

4. Proof of Proposition 1.4 We put

F(x, y) :=

{

x+x^q+y^q+1+∑ν−1

i=0 x²ⁱy^q+1−2i+1 ifp= 2 and q= 2^ν, 2x+ 2x^q−y^q+1−(y²−4x)^q+12 ifpis odd,

that is, F is obtained from the homogeneous polynomial in Proposition 1.4 by puttingx₀= 1, x₁=x, x₂ =y. Since the polynomialF is of degreeq+ 1 and the plane curveB is also of degreeq+ 1, it is enough to show thatF(t^q+1, t^q+t) = 0.

Suppose thatp= 2 andq= 2^ν. We put S(x, y) :=

ν−1

∑

i=0

(x y²

)2ⁱ

.

ThenS(x, y) is a root of the Artin-Schreier equation s²+s=

(x y²

)q

+ x y².

HenceS1:=S(t^q+1, t^q+t) is a root of the equations²+s=b, where b:=

[ t^q+1 (t^q+t)²

]q

+ t^q+1

(t^q+t)² = t^2q2+q+1+t^q2+3q+t^q2+q+2+t^3q+1 (t^q+t)^2q+2 . We put

S^′(x, y) := x+x^q+y^q+1 y^q+1 .

We can verify that S₂:=S^′(t^q+1, t^q+t) is also a root of the equations²+s=b.

Hence we have either S₁=S₂ orS₁ =S₂+ 1. We can easily see that both of the rational functionsS₁andS₂onP¹have zero att=∞. HenceS₁=S₂ holds, from which we obtainF(t^q+1, t^q+t) = 0.

Suppose thatpis odd. We put

S(x, y) := 2x+ 2x^q−y^q+1, S1:=S(t^q+1, t^q+t), and S^′(x, y) := (y²−4x)^q+12 , S2:=S^′(t^q+1, t^q+t).

Then it is easy to verify that both ofS²₁ andS₂² are equal to

t^2q2+2q−2t^2q2+q+1+t^2q2+2−2t^q2+3q+ 4t^q2+2q+1−2t^q2+q+2+t^4q−2t^3q+1+t^2q+2. Therefore either S₁ = S₂ or S₁ = −S₂ holds. Comparing the coefficients of the top-degree terms of the polynomials S₁ and S₂ of t, we see that S₁ =S₂, whence F(t^q+1, t^q+t) = 0 follows.

5. Proof of Propositions 1.6 and 1.8 We consider the universal family

L:={(P, Q)∈P¹×P² | Q∈l_P } of the lineslP, which is defined by

x−t^qy+t^2q = 0 inP¹×P², and let

π1:L→P¹, π2:L→P² be the projections. We see thatπ1:L→P¹ has two sections

σ1 : t7→(t, x, y) = (t, t^q+1, t^q+t), σ_q : t7→(t, x, y) = (t, t^q2+q, t^q2+t^q).

ForP ∈P¹, we haveπ2(σ1(P)) =φ(P) andlP ∩B={π2(σ1(P)), π2(σq(P))}. Let Σ1 ⊂L and Σq ⊂L denote the images of σ1 and σq, respectively. Then Σ1 and Σq are smooth curves, and they intersect transversely. Moreover, their intersection points are contained inπ₁⁻¹(P¹(Fq²)).

We denote byM the fiber product ofγ:Sd→P²andπ2:L→P²overP². The pull-backπ₂^∗B ofB byπ2 is equal to the divisorqΣ1+ Σq. HenceM is defined by (5.1)

{

z^d= (y−t^q−t)^q(y−t^q2−t^q), x−t^qy+t^2q= 0.

We denote byM →M the normalization, and by α:M →L, η:M →Sd

the natural projections. Since d is prime to q, the cyclic covering α: M → L of degreed branches exactly along the curve Σ₁∪Σ_q. Moreover, the singular locus Sing(M) ofM is located over Σ₁∩Σ_q, and hence is contained inα⁻¹(π⁻₁¹(P¹(Fq²))).

Sinceη is dominant andρ: ˜Sd→Sdis birational, η induces a rational map η^′:M· · · →S˜d.

LetAdenote the affine open curveP¹\P¹(Fq²). We put L_A:=π⁻₁¹(A), M_A:=α⁻¹(L_A).

Note thatMAis smooth. Letπ1,A:LA→AandαA:MA→LAbe the restrictions of π1 and α, respectively. IfP ∈A, then lP is disjoint from Sing(B), and hence η(α⁻¹(π⁻₁¹(P))) = γ⁻¹(lP) is disjoint from Sing(Sd). Therefore the restriction of η^′ toMA is a morphism. It follows that we have a proper birational morphism

β : ˜M →M

from a smooth surface ˜M toM such thatβinduces an isomorphism fromβ⁻¹(MA) toMA and that the rational mapη^′ extends to a morphism ˜η: ˜M →S˜d. Summing up, we obtain the following commutative diagram:

(5.2)

M_A ,→ M˜ −→^η˜ S˜_d

|| ¤ ↓^β ↓^ρ M_A ,→ M −→^η S_d

αA↓ ¤ ↓^α ↓^γ

LA ,→ L −→^π2 P²

π1,A↓ ¤ ↓^π1 A ,→ P¹ .

Since the defining equation x−t^qy+t^2q = 0 ofL in P¹×P² is a polynomial in k[x, y][t^q], and its discriminant as a quadratic equation of t^q is y²−4x ̸= 0, the projection π2 is a finite morphism of degree 2q and its inseparable degree is q. Hence η is also a finite morphism of degree 2q and its inseparable degree is q. Therefore, in order to prove Proposition 1.6, it is enough to show that M is rational. We denote byk(M) =k(M) the function field ofM. Sincex=t^qy−t^2q on M, the fieldk(M) is generated over k byy, z and t. Let c denote the integer (q+ 1)/d, and put

˜

z:= z

(y−t^q−t)^c ∈k(M).

Then, from the defining equation (5.1) ofM, we have

˜

z^d =y−t^q2−t^q y−t^q−t . Therefore we have

y= z˜^d(t^q+t)−(t^q2+t^q)

˜

z^d−1 ,

and hencek(M) is equal to the purely transcendental extensionk(˜z, t) ofk. Thus Proposition 1.6 is proved.

We put

Ξ := ˜M\MA=β⁻¹(α⁻¹(π₁⁻¹(P¹(Fq²)))).

Since the cyclic covering α : M → L branches along the curve Σ1 =σ1(P¹), the sectionσ1:P¹ →Lof π1 lifts to a section ˜σ1:P¹ →M ofπ1◦α. Let ˜Σ1 denote the strict transform of the image of ˜σ1 byβ : ˜M →M.

Lemma 5.1. The Picard groupPic( ˜M)ofM˜ is generated by the classes ofΣ˜1and the irreducible components of Ξ.

Proof. Since Σ1∩Σq∩LA=∅, the morphism π1,A◦αA:MA→A

is a smoothP¹-bundle. LetDbe an irreducible curve on ˜M, and letebe the degree of

π1◦α◦β|D:D→P¹.

Then the divisorD−eΣ˜1on ˜M is of degree 0 on the general fiber of the smoothP¹- bundleπ1,A◦αA. Therefore (D−eΣ˜1)|M_Ais linearly equivalent inMAto a multiple of a fiber ofπ1,A◦αA. HenceDis linearly equivalent to a linear combination of ˜Σ1

and irreducible curves in the boundary Ξ = ˜M \MA. ¤ The rational curves on ˜Sd listed in Proposition 1.8 are exactly equal to the irreducible components of

ρ⁻¹(γ⁻¹( ∪

P∈P¹(Fq2)

lP)).

LetV ⊂H²( ˜Sd) denote the linear subspace spanned by the classes of these rational curves. We will show thatV =H²( ˜Sd).

Leth∈H²( ˜Sd) denote the class of the pull-back of a line ofP²by the morphism γ◦ρ : ˜Sd → P². Suppose that P ∈ P¹(Fq). Then lP is disjoint from Sing(B).

Therefore we have

h= [(γ◦ρ)^∗(lP)] = [l⁽⁰⁾_P ] +· · ·+ [l^(d_P⁻¹⁾]∈V.

Let ˜Bdenote the strict transform ofBbyγ◦ρ. Then ˜Bis written asd·R, whereR is a reduced curve on ˜S_d whose support is equal to ˜η( ˜Σ₁). On the other hand, the class of the total transform (γ◦ρ)^∗B ofB byγ◦ρis equal to (q+ 1)h. Since the difference of the divisorsd·R and (γ◦ρ)^∗B is a linear combination of exceptional curves ofρ, we have

(5.3) η˜_∗([ ˜Σ₁])∈V.

By the commutativity of the diagram (5.2), we have

˜

η(Ξ) ⊂ ρ⁻¹(γ⁻¹( ∪

P∈P¹(Fq2)

lP)).

Hence, for any irreducible component Γ of Ξ, we have

(5.4) η˜_∗([Γ])∈V.

LetC be an arbitrary irreducible curve on ˜S_d. Then we have

˜

η_∗η˜^∗([C]) = 2q[C].

By Lemma 5.1, there exist integersa,b₁, . . . , b_mand irreducible components Γ₁, . . . ,Γ_m of Ξ such that the divisorη^∗C of ˜M is linearly equivalent to

aΣ˜1+b1Γ1+· · ·+bmΓm. By (5.3) and (5.4), we obtain

[C] = 1

2qη˜_∗η˜^∗([C])∈V.

ThereforeV ⊂H²( ˜S_d) is equal to the linear subspace spanned by the classes of all curves. Combining this fact with Corollary 1.7, we obtainV =H²( ˜S_d).

6. Supersingular K3 surfaces

In this section, we prove Proposition 1.9. First, we recall some facts on super- singularK3 surfaces. LetY be a supersingularK3 surface in characteristicp, and let NS(Y) denote its N´eron-Severi lattice, which is an even hyperbolic lattice of rank 22. Artin [3] showed that the discriminant of NS(Y) is written as−p^2σ, where σ is a positive integer ≤ 10. This integer σ is called the Artin invariant of Y. Ogus [13, 14] and Rudakov-Shafarevic [15] proved that, for eachp, a supersingular K3 surface with Artin invariant 1 is unique up to isomorphisms. Let X_p denote the supersingularK3 surface with Artin invariant 1 in characteristicp. It is known that X₃ is isomorphic to the Fermat quartic surface, and that X₅ is isomorphic to the Fermat sextic double plane. (See, for example, [12] and [17], respectively.) Therefore, in order to prove Proposition 1.9, it is enough to prove the following:

Proposition 6.1. Suppose that(d, q+1) = (4,4)or(2,6). Then, among the curves on S˜_d listed in Proposition 1.8, there exist 22 curves whose classes together with the intersection pairing form a lattice of rank22 with discriminant−p².

Proof. Suppose that p = q = 3 and d = 4. We put α := √

−1 ∈ F9, so that F9 := F3(α). Consider the projective space P³ with homogeneous coordinates [w:x0:x1:x2]. By Proposition 1.4, the surfaceS4is defined inP³ by an equation

w⁴= 2(x³₀x1+x0x³₁)−x⁴₂−(x²₂−x1x0)². Hence the singular locus Sing(S4) ofS4consists of the three points

Q0 := [0 : 1 : 1 : 0] (located over φ([1 :α]) =φ([1 :−α])∈B), Q1 := [0 : 1 : 2 : 1] (located over φ([1 : 1 +α]) =φ([1 : 1−α])∈B), Q₂ := [0 : 1 : 2 : 2] (located over φ([1 : 2 +α]) =φ([1 : 2−α])∈B), and they are rational double points of typeA₃. The minimal resolutionρ: ˜S₄→S₄ is obtained by blowing up twice over each singular pointQ_a (a∈F3). The rational

curvesl_P⁽ⁱ⁾ on ˜S₄ given in Proposition 1.8 are the strict transforms of the following 40 lines ¯L^(ν)τ inP³contained inS4, where ν= 0, . . . ,3:

L¯^(ν)₀ := {x₁=w−α^νx₂= 0},

L¯^(ν)₁ := {x0+x1−x2=w−α^ν(x2+x0) = 0}, L¯^(ν)₂ := {x₀+x₁+x₂=w−α^ν(x₂−x₀) = 0}, L¯^(ν)_∞ := {x₀=w−α^νx₂= 0},

L¯^(ν)_±α := {−x0+x1±αx2=w−α^νx2= 0},

L¯^(ν)_1±α := {±αx₀+x₁+ (−1±α)x₂=w−α^ν(x₂+x₀) = 0}, L¯^(ν)_2±α := {∓αx0+x1+ (1±α)x2=w−α^ν(x2−x0) = 0}.

We denote byL^(ν)τ the strict transform of ¯L^(ν)τ byρ. Note that the image of ¯L^(ν)τ by the covering morphismS4→P² is the linelφ([1:τ]). Note also that, ifτ∈F3∪ {∞}, then ¯L^(ν)τ is disjoint from Sing(S4), while ifτ =a+bα∈F9\F3 witha∈F3 and b ∈F3\ {0} ={±1}, then ¯L^(ν)τ ∩Sing(S4) consists of a single pointQa. Looking at the minimal resolution ρover Qa explicitly, we see that the three exceptional (−2)-curves in ˜S4overQa can be labeled asEa−α, Ea, Ea+αin such a way that the following hold:

• 〈Ea−α, Ea〉=〈Ea, Ea+α〉= 1,〈Ea−α, Ea+α〉= 0.

• Suppose thatb∈ {±1}. ThenL^(ν)_a+bαintersectsEa+bα, and is disjoint from the other two irreducible componentsEa and Ea−bα.

• The four intersection points ofL^(ν)_a+bα(ν = 0, . . . ,3) andE_a+bαare distinct.

Using these, we can calculate the intersection numbers among the 9 + 40 curvesEτ

andL^(ν)_τ′ (τ ∈F9, τ^′∈F9∪ {∞}, ν = 0, . . . ,3). From among them, we choose the following 22 curves:

E₋α, E0, Eα, E1−α, E1, E1+α, E2−α, E2, E2+α, L⁽⁰⁾₀ , L⁽¹⁾₀ , L⁽²⁾₀ , L⁽³⁾₀ , L⁽⁰⁾₁ , L⁽¹⁾₁ , L⁽⁰⁾₂ , L⁽¹⁾₂ , L⁽¹⁾_∞, L⁽⁰⁾_−α, L⁽¹⁾_−α, L⁽²⁾_1−α, L⁽⁰⁾_2−α.

Their intersection numbers are calculated as in Table 6.1. We can easily check that this matrix is of determinant−9.

The proof for the case p = q = 5 and d = 2 is similar. We put α := √ 2 so that F25=F5(α). In the weighted projective spaceP(3,1,1,1) with homogeneous coordinates [w:x0:x1:x2], the surfaceS2 forp=q= 5 is defined by

w²= 2(x⁵₀x₁+x₀x⁵₁)−x⁶₂−(x²₂+x₀x₁)³. The singular locus Sing(S2) consists of ten ordinary nodes

Q_{{a+bα,a−bα}} (a∈F5, b∈ {1,2})

located over the nodes φ([1 : a+bα]) = φ([1 : a−bα]) of the branch curve B.

Let E_{{a+bα,a−bα}} denote the exceptional (−2)-curve in ˜S₂ over Q_{{a+bα,a−bα}} by the minimal resolution. As the 22 curves, we choose the following eight exceptional

2 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 4

−2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 −2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 −2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 −2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 −2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 −2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 −2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −2 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 −2 1 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 −2 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 −2 0 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 −2 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 −2 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 −2 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 −2 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 −2 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 −2 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 −2 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 −2 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 1 0 0 0 1 0 −2

3 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 5

Table 6.1. Gram matrix of NS( ˜S4) forq= 3

2 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 4

−2 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 −2 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 −2 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 −2 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 −2 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 −2 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 −2 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 −2 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 −2 3 1 1 0 1 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 3 −2 0 0 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 −2 0 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 −2 0 0 1 1 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 −2 0 0 0 1 1 1 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 −2 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 1 1 0 0 −2 1 1 0 1 0 1 1 0 0 0 1 0 0 0 0 0 1 1 1 0 1 1 −2 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 1 −2 1 1 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 0 0 0 1 −2 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 1 0 −2 0 1 0 0 0 0 0 0 1 0 0 1 0 0 1 1 0 0 1 1 0 0 −2 1 1 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 0 1 1 −2 1 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 1 1 −2

3 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 5

Table 6.2. Gram matrix of NS( ˜S₂) forq= 5

(−2)-curves

E_{−α,α}, E_{−2α,2α}, E_{{1−α,1+α}}, E_{{1−2α,1+2α}}, E_{{2−α,2+α}}, E_{{3−2α,3+2α}}, E_{{4−α,4+α}}, E_{{4−2α,4+2α}}, and the strict transforms of the following 14 curves onS2:

{ x1 = w−2αx23 = 0 }, { x₁ = w+ 2αx₂³ = 0 },

{ x₀+x₁+ 4x₂ = w+ 2α(3x₀+x₂)³ = 0 }, { 3x0+x1+ 3αx2 = w−2αx23 = 0 }, { 2x0+x1+ 4αx2 = w+ 2αx23 = 0 }, { 3x₀+x₁+ 2αx₂+ 3x₀ = w−2αx₂³ = 0 },

{ (3 + 3α)x₀+x₁+ (4 +α)x₂ = w+ 2α(3x₀+x₂)³ = 0 }, { (4 +α)x₀+x₁+ (4 + 2α)x₂ = w+ 2α(3x₀+x₂)³ = 0 }, { (2 + 3α)x0+x1+ (3 + 3α)x2 = w−2α(x0+x2)³ = 0 }, { (1 +α)x0+x1+ (3 +α)x2 = w−2α(x0+x2)³ = 0 }, { (1 +α)x0+x1+ (2 + 4α)x2 = w−2α(x2+ 4x0)³ = 0 }, { (2 + 3α)x0+x1+ (2 + 2α)x2 = w+ 2α(x2+ 4x0)³ = 0 }, { (3 + 3α)x0+x1+ (1 + 4α)x2 = w−2α(x2+ 2x0)³ = 0 }, { (4 + 4α)x0+x1+ (1 + 2α)x2 = w−2α(x2+ 2x0)³ = 0 }. Their intersection matrix is given in Table 6.2. It is of determinant−25. ¤ Remark6.2. In the caseq= 5, the Ballico-Hefez curveB is one of the sextic plane curves studied classically by Coble [5].

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Department of Mathematics, Graduate School of Science, Hiroshima University, 1- 3-1 Kagamiyama, Higashi-Hiroshima, 739-8526 JAPAN

E-mail address:hoangthanh2127@yahoo.com

Department of Mathematics, Graduate School of Science, Hiroshima University, 1- 3-1 Kagamiyama, Higashi-Hiroshima, 739-8526 JAPAN

E-mail address:shimada@math.sci.hiroshima-u.ac.jp